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\section{Discussion}

Our calibration dataset (dataset \#1) contained detailed volume measurements for 8192 trees from 19 species. Many references about expansion factors found in the literature were based on volume and/or biomass estimates obtained through allometric equations and thus were not directly based on field measurements \citep[e.g.,][]{Schroeder1997,Lehtonen2004,Jalkanen2005}, especially when the studies were oriented toward large scale estimates. When $BEF$s were estimated directly from field measurements, relatively few species were generally considered \citep[e.g.,][]{Pajtik2008,Pajtik2011,Skovsgaard2012} or the range of tree size was limited (see some examples of studies below) and consequently the equations were probably not suitable for other species and for bigger trees. Modeling $VEF$ directly from field measurements enabled to avoid propagation of modeling errors. Working with a large range of tree species was important for the generic aspect of our model.

The data used for calibrating the models were collected between 1920 and 1985. It is possible that the tree shape has changed since these measurements, mainly due to changes in silvicultural practices and less likely due to the climate change effect. It was not possible to investigate that point with our dataset. 

The major weakness of the calibration dataset is that it represents at 90\% pure, even-aged high forests \citep{Vallet2006} and rather small and medium trees in diameter compared with the French National Forest Inventory data (not shown). 
It is important to remind that the dataset \#2 used for validation was specially chosen (see Section \ref{M_and_M}) to complement the available data with under-represented species, regions or $DBH$ values, which often led to use the model in extrapolation:

\begin{itemize}
\item{For \textit{Fraxinus} species, the model was calibrated with trees ranging between 8.3 and 43.4 cm in diameter (dataset \#1) whereas five trees from dataset \#2 had $DBH$ ranging between 44.7 and 64.9 cm. The model was therefore used in extrapolation for these trees leading to big errors both in terms on $VEF$ and $Vtot$.} 
\item{For \textit{Pseudotsuga}, the trees from the calibration dataset were young with a maximum $DBH$ of 34.1 cm since the species was recently introduced in France at the date of measurements. It would therefore be preferable to use the generic gymnosperm model (i.e., fixed-effects model and $G = 0$) for this species.}
\item{For \textit{Fagus} and \textit{Quercus} species, the model was calibrated mostly on high forest trees. The big trees of dataset \#2 coming from coppice-with-standards stands tended to have underestimated values of $VEF$, except for one declining tree.}
\end{itemize}

The predictions obtained for trees consistent with the calibration dataset \#1 (i.e., issued from even-aged high forests, with a comparable range of diameters, not forked trees) were satisfactory in term of RMSE (values of the same order). To go further, we have tested a calibration of the model based on dataset \#1 and \#2 pooled together since the two datasets were complementary. However, dataset \#2 was two small in comparison with dataset \#1 to impact significantly the parameter estimates. In the model with $DBH$ only it was observed that the slopes were slightly increasing for \textit{Fraxinus} and for \textit{Quercus}, which tended to improve the model for big trees.  

For genera represented by a large number of trees like \textit{Pinus}, \textit{Fagus} or \textit{Quercus}, a quite high variability of $VEF$ was observed. For \textit{Pinus} genus, this variability could result mainly from differences in origin of the trees (e.g., region, site) and differences between species. For angiosperm genera, it could be explained in addition by a larger range of silviculture practices (e.g., coppice-with-standards, pure even-aged high forests, mixed-species forests). \cite{VanCamp2004} who have compared several $BEF$s published in Europe \citep{Lowe2000} and their accuracy for predicting total above-ground biomass for oak, beech and ash concluded that there was a need to refine $BEF$ for oak due to a high variability observed in the oak population.

It was not possible to get easily a reliable information about the silviculture for each tree of the dataset \#1 and therefore to take this aspect into account in our models. The silvicultural effect was at least partially, but not perfectly, taken into account in our model through the variable $\frac{DBH}{H^{2}}$. This variable is also related to the \textit{hardiness} used by \cite{Vallet2006} to model $Vtot$ directly from circumference at breast height ($C130$) and height. The variable $\frac{\sqrt{C130}}{H}$ was called \textit{hardiness} by \cite{Vallet2006} because the higher it is, the more cone-shaped is the tree, and the smaller it is, the slenderer is the tree. And it is known that the \textit{slenderness} of a tree, defined as $\frac{H}{DBH}$, is related to the silviculture and in particular to stand density \citep{Wang1998,Jagodzinski2009}. In addition, the \textit{hardiness} was relatively independent of the $DBH$ and therefore added new information in comparison with $DBH$ \citep{Vallet2006}. 

The presence of a fork also had a big impact on $VEF$. For a given $DBH$, a forked tree had a greater $VEF$ and a greater total woody volume than a not forked tree. This was observed in our study for \textit{Quercus robur/petraea} and \textit{Quercus pubescens}. \cite{Adu-Bredu2008} have shown for teak trees that the stem volume was decreasing with the number of forks and this suggested that forked trees, if they have in addition a greater total volume, have consequently much more volume in their branches and greater $VEF$s.

Regarding the parameter estimates, the model seemed to be consistent at several levels. The value of 6.830 for $\beta_1$ (6.438 in the model with $DBH$ only), which controlled the curve location along the X-axis, was in the order of the lower diameter limit of 7 cm that was fixed in the study. The error variance power values $\delta_{G_{i}}$ obtained for angiosperms and gymnosperms were negative, which was consistent with an increasing of the $VEF$ variability for small diameter trees. It was shown that the variability was higher for angiosperms than for gymnosperms, which reflected a greater variability in tree architecture and volume distribution for angiosperms. This could be related to a greater control of the volume allocation in gymnosperms, for which the growth is largely dominated by the stem \citep{Barthelemy2007}, and maybe also to more complex silvicultural practices for angiosperms than for gymnosperms (e.g., coppice-with-standards vs. high forests, mixed-species stands vs. pure stands). The slopes of the relationship between $VEF$ and $DBH$ were higher for angiosperms than for gymnosperms indicating that, for a given diameter at breast height, angiosperms had much more volume in their branches in comparison with the corresponding stem volume.  

The differences observed between species for the relationship between $VEF$ and $DBH$ probably reflected differences in tree architecture and possibly in the silviculture traditionally associated with the considered species. Regarding our results and in particular Fig. \ref{Predicted_VEF_vs_genus}, the approximate value of 1.5 given by \cite{Pretzsch2009} seems appropriate for angiosperms but not for gymnosperms for which the $VEF$ and $Vtot$ would be highly overestimated. \cite{IPCC2003} gives an average $BEF$ of approximately 2 and \cite{FAO2005} estimates an average $BEF$ of 2.2. However, these $BEF$s generally developed at the stand scale integrate data from numerous species of different countries, and therefore are not directly comparable with our results. 
Some groups of species appeared like \textit{Picea} and \textit{Abies} or like \textit{Fraxinus} and \textit{Fagus} and in a further study it would be interesting to link these groups to architectural and ecological traits known for these species.
The variability that was observed within the genera \textit{Pinus} and \textit{Quercus} raises the question of developing models at the species level rather than at the genus level, at least for atypical species like \textit{Quercus ilex} or for species sufficiently well represented in our dataset.

Direct comparison of our results to the literature is not easy since various definitions of $VEF$, $BEF$ or $BCEF$ were used. However, $VEF$ and $BEF$ are quite equal and databases and publications exist that provide wood basic density for numerous temperate tree species \citep[e.g.,][]{Chave2009,Zanne2009}, which could be used to convert easily $VEF$ to $BCEF$. Regarding the relationship with $DBH$, the trends should be comparable for $VEF$ and $BCEF$ since multiplying by a mean wood density would not change the sign of the slopes.
In our model the slope of the relationship was constrained to be positive for trees above a given diameter depending on the model parameters. In the literature, the slopes obtained for $BEF$ were often negative \citep{Somogyi2007,Pajtik2008,Pajtik2011,Sanquetta2011} because the definitions used for expansion factors were different or because the trees were still young or small and only the first part of the relationship (i.e., the decreasing part) was obtained. 
\cite{Sanquetta2011} showed aboveground biomass over stem biomass ratio decreasing with $DBH$ up to 20 cm in diameter and then staying approximately constant between 1.0 and 1.5 up to 40 cm in diameter for \textit{Pinus} in Brazil. Due to the variability that we observed for the genus \textit{Pinus}, our results could be considered as rather consistent with this study.
\cite{Somogyi2007} showed in their Fig. 2 a stem volume over merchantable volume (stem and branches greater than 7 cm in diameter) ratio as a function of $DBH$ for beech, oak, pine and spruce. This volume ratio corresponded approximately to the inverse of our $VEF$. Considering this, our results were consistent with the plots of Somogyi et al. with a $VEF$ increasing with $DBH$ and with lower slopes for spruce and pine than for oak and beech. In their Fig. 3, Somogyi et al. showed total volume over stem volume (stem part greater than 5 cm in diameter) ratio decreasing with $DBH$ for \textit{Robinia pseudoacacia}, with $DBH$ ranging from 5 to 15 cm, whereas in our study and for the same species $DBH$ varied between 15 and 24 cm. \cite{Pajtik2008} showed a total aboveground biomass over stem biomass ratio decreasing with stem base diameter for \textit{Picea abies} and diameters up to approximately 7 cm, whereas in our study $DBH$ for the same species ranged between 8 and 67 cm. \cite{Pajtik2011} showed total aboveground biomass over stem volume ratio decreasing with stem base diameter for beech, pine, oak and spruce and diameters up to approximately 8 cm, whereas in our study we considered bigger diameters for all these species. Previous cited studies emphasize the lack of studies based on medium and big trees and the fact that our dataset including bigger trees was particularly valuable.

Plots of $BEF$ as a function of growing stock at the stand level (in m$^3$/ha) show negative or almost null slopes as well \citep{Brown2002,Lehtonen2004,FAO2005,Teobaldelli2009,Guo2010}. However, these studies are difficult to compare directly with others because relationships are studied at the stand level and not at the individual tree level and the definitions of compartments "from what" and "to what" conversions are done may differ between studies \citep{Teobaldelli2009}.





